We study models with fractonlike order based on ${\mathbb{Z}}_{2}$ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy ${S}_{A}$ of these models in a subregion $A$ of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to $A$.
We study models with fractonlike order based on Z2 lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy SA of these models in a subregion A of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to A.
We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields ZN gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile ``fracton'' excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub ``twisted fracton models.'' A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a 3-torus and computing their ground-state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group G, a choice of an Abelian normal subgroup N, and a choice of foliation structure. These hybrid fracton orders, examples of which were introduced in a previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional G gauge theory and a pure fracton order, by varying the choice of normal subgroup N. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of G and N, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.
Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called Fp gauge theories, in which gauge transformations of rank-k fields are specified by rank-(k−p) gauge parameters. The Z2 rank-two theory of type F2, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1) rank-two theory of type F2 is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this F2 theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to F2 theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.